Chapter 4 Applications and solutions of linear programming, after completing this chapter, you should be able to: Formulate linear programming problems with different objectives, set up constraints that have unique structures, translate statements into constraint formulas, utilize excel to solve a variety of linear programming problems,...
Introduction to Management Science with Spreadsheets Stevenson and Ozgur First Edition Part Introduction to Management Science and Forecasting Chapter 4 Applications and Solutions of Linear Programming McGrawHill/Irwin Copyright © 2007 by The McGrawHill Companies, Inc. All rights reserved Learning Objectives After completing this chapter, you should be able to: Formulate linear programming problems with different objectives Set up constraints that have unique structures Translate statements into constraint formulas Utilize Excel to solve a variety of linear programming problems Interpret the Excel output of linear programming problems Make managerial conclusions based on computer (Excel) output Explain at least two applications of linear programming Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–2 Elements Elements of of Linear Linear Programming Programming • Issues in formulating problems – Degree of simplification of the real-world problem – Violation of the assumptions of linear programming (additivity, divisibility, certainty, and proportionality) • Linear programming problems may include: – Different types of decision variables definitions – Different types of constraints – Different types of objective functions Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–3 Example Example4-1 4-1Product-Mix Product-MixProblem Problem The Style and Comfort Furniture Manufacturing Company wishes to determine its production schedule for the next quarter The company produces four types of furniture, including sofas, love seats, recliners, and coffee tables The profit contribution from selling one sofa is $120, one love seat is $105, one recliner is $150, and one coffee table is $73 The quarterly production budget is set at $180,000 Each unit of a sofa, love seat, recliner, and coffee table costs $400, $300, $500, and $150, respectively The sales forecasts indicate that the potential sales volume is limited to 200 units of sofas, 150 units of love seats, 100 units of recliners, and 400 units of coffee tables There are an aggregate of 800 machine hours available and 1,200 labor hours available Table 4-1 summarizes the number of machine hours and the number of labor hours required per unit of each product Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–4 Table Table4–1 4–1 Per PerUnit UnitMachine Machineand andLabor LaborHour HourRequired Requiredfor forEach EachProduct Product Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–5 Style Style of of Comfort Comfort Furniture Furniture Example Example Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–6 Exhibit Exhibit4-1 4-1 Input InputScreen Screenfor forthe theStyle Styleand andComfort ComfortFurniture FurnitureCompany CompanyProblem Problem Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–7 Exhibit Exhibit4–2 4–2 Solver SolverDialog DialogBox Box Exhibit Exhibit44–3 –3 Add AddConstraint ConstraintDialog DialogBox Box Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–8 Exhibit Exhibit4–4 4–4 Completed CompletedSolver SolverDialog DialogBox Box There are three types of Solver cells: Target cell shows the overall performance measure (i.e., the maximum profit or the minimum cost) Output cell shows the output or quantity of the decision variables based on the changing cells Data cells simply show the data of the problem (constraint coefficients, objective function coefficients, and right-hand-side constraints) Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–9 Exhibit Exhibit4-5 4-5 Solver SolverOptions OptionsDialog DialogBox Box Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–10 Example Example4-7 4-7Financial FinancialPlanning Planning(cont’d) (cont’d) Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–29 Example Example4-7 4-7Financial FinancialPlanning Planning(cont’d) (cont’d) Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–30 Example Example4-8 4-8Portfolio PortfolioSelection Selection A conservative investor has $100,000 to invest The investor has decided to use three vehicles for generating income: municipal bonds, a certificate of deposit (CD), and a money market account After reading a financial newsletter, the investor has also identified several additional restrictions on the investments: No more than 40 percent of the investment should be in bonds The proportion allocated to the money market account should be at least double the amount in the CD The annual return will be percent for bonds, percent for the CD, and percent for the money market account Assume the entire amount will be invested Formulate the LP model for this problem, ignoring any transaction costs and the potential for different investment lives Assume that the investor wants to maximize the total annual return Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–31 Example Example4-8 4-8Portfolio PortfolioSelection Selection(cont’d) (cont’d) Copyrightâ2007TheMcGrawưHill Companies.Allrightsreserved. McGrawư Hill/Irwin432 Production Production Applications Applications ã Linear programming in production management in manufacturing: – Multiperiod production scheduling – Workforce scheduling – Make-or-buy decisions Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–33 Example Example4-9 4-9 Multiperiod MultiperiodProduction ProductionScheduling Scheduling Morton and Monson Inc is a small manufacturer of parts for the aerospace industry The production capacity for the next four months is given as follows: Month January February March April Production Capacity in Units Regular Production Overtime Production 3,000 500 2,000 400 3,000 600 3,500 800 The regular cost of production is $500 per unit and the cost of overtime production is $150 per unit in addition to the regular cost of production The company can utilize inventories to reduce fluctuations in production, but carrying one unit of inventory costs the company $40 per unit per month Currently there are no units in inventory However, the company wants to maintain a minimum safety stock of 100 units of inventory during the months of January, February, and March The estimated demand for the next four months is as follows: Month January Demand 2,800 February 3,000 March 3,500 April 3,000 Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–34 Example Example4-9 4-9 Multiperiod MultiperiodProduction ProductionScheduling Scheduling(cont’d) (cont’d) Copyright © 2007 The McGrawHill Continued on next slide Companies. All rights reserved. McGraw Hill/Irwin 4–35 Example Example4-9 4-9 Multiperiod MultiperiodProduction ProductionScheduling Scheduling(cont’d) (cont’d) Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–36 Example Example4-9 4-9 Multiperiod MultiperiodProduction ProductionScheduling Scheduling(cont’d) (cont’d) Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–37 Example Example4-10 4-10 Workforce WorkforceScheduling Scheduling Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–38 Example Example4-11 4-11 Make-or-Buy Make-or-BuyDecisions Decisions Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–39 Example Example4-12 4-12 Agriculture AgricultureApplications Applications A farm owner in Des Moines, Iowa, is interested in determining how to divide the farmland among four different types of crops The farmer owns two farms in separate locations and has decided to plant the following four types of crops in these farms: corn, wheat, bean, and cotton The first farm consists of 1,450 acres of land, while the second farm consists of 850 acres of land Any of the four crops may be planted on either farm However, after a survey of the land, based on the characteristics of the farmlands, Table 4-7 shows the maximum acreage restrictions the farmer has placed for each crop Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–40 Data Data Envelopment Envelopment Analysis Analysis • Data envelopment analysis (DEA) – The use linear programming to measure the relative efficiency of the departments within an organization or relative efficiencies of two different organizations • Fast food restaurants, hospitals, schools, and banks – Measuring the efficiency of departments within the same organization is difficult because departments have multiple input measures (number of workers, cost of labor, cost of machine operations, pay scale for employees, and cost of advertising) and multiple output measures (profit, sales, and market share) Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–41 Example Example4-12 4-12 Airline AirlineEfficiency Efficiency A consumer advocacy group would like to evaluate the efficiency of three airline companies with respect to several criteria that are important to consumers The three companies are Beta, Unified, and Southeast The output measures are identified as average minutes late arriving, number of luggage loser per 10,000 customers, waiting time for check-in, friendliness, and quality of air service (result of a questionnaire that measured this output on a 100point scale) The input measures were labor cost per day, number of flights per day, and over head cost per day Table 4-9 summarizes the results Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–42 Other Other Applications Applications • Transportation problems – Developing distribution plans that will minimize total distribution costs given the capacities of the various factories and the needs of the warehouses • Assignment problems – Assigning jobs to machines in such a way that the total cost of performing the jobs is minimized • Project management problems – Finding the minimum project duration for project competition to shorten various project tasks and minimize total project costs Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin 4–43 ... Hill/Irwin ? ?4? ??23 Example Example 4-6 4- 6(cont’d) (cont’d) Problem formulation Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin ? ?4? ?? 24 Example Example 4-6 4- 6(cont’d)... McGraw Hill/Irwin ? ?4? ??12 Example Example 4-2 4- 2Diet DietProblem Problem Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin ? ?4? ??13 Example Example 4-3 4- 3Blending BlendingProblem... Hill/Irwin ? ?4? ??37 Example Example 4-1 0 4- 10 Workforce WorkforceScheduling Scheduling Copyright © 2007 The McGrawHill Companies. All rights reserved. McGraw Hill/Irwin ? ?4? ??38 Example Example 4-1 1 4- 11